Torus Topology/Mathematical Notation

Mathematical Notation This appendix provides a reference for the mathematical notation used throughout this book on torus topology. General Notation 1. - The standard 2-dimensional torus 2. - The n-dimensional torus 3. - The torus as a product of circles 4. - The torus as a quotient space Coordinate Systems 1. - Angular coordinates on the torus, where represents the poloidal angle and represents the toroidal angle 2. - Parametric coordinates, typically with domain 3. - Cartesian x-coordinate of the torus 4. - Cartesian y-coordinate of the torus 5. - Cartesian z-coordinate of the torus Topological Invariants 1. - Fundamental group of the torus 2. - First homology group 3. - Euler characteristic of the torus 4. - Genus of the torus Differential Geometry 1. - Gaussian curvature 2. - Gaussian curvature at a point 3. - First fundamental form (metric tensor) 4. - Covariant derivative of vector field Y with respect to X 5. - A geodesic curve on the torus Algebraic Topology 1. - The k-th cohomology group of the torus 2. - A differential form on the torus 3. - Integration of a 2-form over the torus 4. - Commutator of elements in the fundamental group Torus Knots 1. - The (p,q)-torus knot 2. - Parametrization of a (p,q)-torus knot 3. - Linking number between two knots on the torus Computational Aspects 1. - Discretized torus with mesh size h 2. - Discrete Laplacian on the torus 3. - Discrete approximation of a function f on the torus 4. - Error norm between exact and approximate solutions